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Theorem bnj873 32095
Description: Technical lemma for bnj69 32179. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj873.4 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
bnj873.7 (𝜑′[𝑔 / 𝑓]𝜑)
bnj873.8 (𝜓′[𝑔 / 𝑓]𝜓)
Assertion
Ref Expression
bnj873 𝐵 = {𝑔 ∣ ∃𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′)}
Distinct variable groups:   𝐷,𝑓,𝑔   𝑓,𝑛,𝑔   𝜑,𝑔   𝜓,𝑔
Allowed substitution hints:   𝜑(𝑓,𝑛)   𝜓(𝑓,𝑛)   𝐵(𝑓,𝑔,𝑛)   𝐷(𝑛)   𝜑′(𝑓,𝑔,𝑛)   𝜓′(𝑓,𝑔,𝑛)

Proof of Theorem bnj873
StepHypRef Expression
1 bnj873.4 . 2 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
2 nfv 1906 . . 3 𝑔𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)
3 nfcv 2974 . . . 4 𝑓𝐷
4 nfv 1906 . . . . 5 𝑓 𝑔 Fn 𝑛
5 bnj873.7 . . . . . 6 (𝜑′[𝑔 / 𝑓]𝜑)
6 nfsbc1v 3789 . . . . . 6 𝑓[𝑔 / 𝑓]𝜑
75, 6nfxfr 1844 . . . . 5 𝑓𝜑′
8 bnj873.8 . . . . . 6 (𝜓′[𝑔 / 𝑓]𝜓)
9 nfsbc1v 3789 . . . . . 6 𝑓[𝑔 / 𝑓]𝜓
108, 9nfxfr 1844 . . . . 5 𝑓𝜓′
114, 7, 10nf3an 1893 . . . 4 𝑓(𝑔 Fn 𝑛𝜑′𝜓′)
123, 11nfrex 3306 . . 3 𝑓𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′)
13 fneq1 6437 . . . . 5 (𝑓 = 𝑔 → (𝑓 Fn 𝑛𝑔 Fn 𝑛))
14 sbceq1a 3780 . . . . . 6 (𝑓 = 𝑔 → (𝜑[𝑔 / 𝑓]𝜑))
1514, 5syl6bbr 290 . . . . 5 (𝑓 = 𝑔 → (𝜑𝜑′))
16 sbceq1a 3780 . . . . . 6 (𝑓 = 𝑔 → (𝜓[𝑔 / 𝑓]𝜓))
1716, 8syl6bbr 290 . . . . 5 (𝑓 = 𝑔 → (𝜓𝜓′))
1813, 15, 173anbi123d 1427 . . . 4 (𝑓 = 𝑔 → ((𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑔 Fn 𝑛𝜑′𝜓′)))
1918rexbidv 3294 . . 3 (𝑓 = 𝑔 → (∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ↔ ∃𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′)))
202, 12, 19cbvabw 2887 . 2 {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)} = {𝑔 ∣ ∃𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′)}
211, 20eqtri 2841 1 𝐵 = {𝑔 ∣ ∃𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′)}
Colors of variables: wff setvar class
Syntax hints:  wb 207  w3a 1079   = wceq 1528  {cab 2796  wrex 3136  [wsbc 3769   Fn wfn 6343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-fun 6350  df-fn 6351
This theorem is referenced by:  bnj849  32096  bnj893  32099
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